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Properties

Label 2-888-111.14-c1-0-25
Degree $2$
Conductor $888$
Sign $-0.467 + 0.883i$
Analytic cond. $7.09071$
Root an. cond. $2.66283$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.56 + 0.745i)3-s + (0.212 − 0.791i)5-s + (−0.149 − 0.258i)7-s + (1.88 − 2.33i)9-s − 1.40·11-s + (2.07 + 0.556i)13-s + (0.258 + 1.39i)15-s + (−4.03 + 1.08i)17-s + (−4.30 − 1.15i)19-s + (0.425 + 0.292i)21-s + (−3.40 − 3.40i)23-s + (3.74 + 2.16i)25-s + (−1.21 + 5.05i)27-s + (2.74 − 2.74i)29-s + (−0.659 − 0.659i)31-s + ⋯
L(s)  = 1  + (−0.902 + 0.430i)3-s + (0.0948 − 0.353i)5-s + (−0.0563 − 0.0976i)7-s + (0.629 − 0.776i)9-s − 0.423·11-s + (0.575 + 0.154i)13-s + (0.0666 + 0.360i)15-s + (−0.978 + 0.262i)17-s + (−0.987 − 0.264i)19-s + (0.0929 + 0.0639i)21-s + (−0.709 − 0.709i)23-s + (0.749 + 0.432i)25-s + (−0.234 + 0.972i)27-s + (0.509 − 0.509i)29-s + (−0.118 − 0.118i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.467 + 0.883i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ -0.467 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.266633 - 0.442905i\)
\(L(\frac12)\) \(\approx\) \(0.266633 - 0.442905i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.56 - 0.745i)T \)
37 \( 1 + (3.30 + 5.10i)T \)
good5 \( 1 + (-0.212 + 0.791i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.149 + 0.258i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.40T + 11T^{2} \)
13 \( 1 + (-2.07 - 0.556i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.03 - 1.08i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.30 + 1.15i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.40 + 3.40i)T + 23iT^{2} \)
29 \( 1 + (-2.74 + 2.74i)T - 29iT^{2} \)
31 \( 1 + (0.659 + 0.659i)T + 31iT^{2} \)
41 \( 1 + (5.90 + 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.89 - 4.89i)T - 43iT^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 + (6.63 + 3.83i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.53 + 0.410i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.777 - 2.89i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.51 - 0.876i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.84 + 1.64i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.542iT - 73T^{2} \)
79 \( 1 + (-4.66 - 1.25i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (7.59 + 4.38i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.75 + 6.56i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-7.62 + 7.62i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.12091654901427707089582247206, −8.951123652702966012781810229830, −8.427204377264367967703821418375, −6.99734531388926629674304853346, −6.37878318476669193111717290147, −5.40449861590926444004077887958, −4.55694361023959937255826209324, −3.70372165890640520638439851559, −2.00946351973166049248603609021, −0.27465360996505013570797290308, 1.55223738664830205087811704021, 2.86027256540030316656287319869, 4.32334963600483524339471615747, 5.20840974368656032228448132980, 6.30509288339090046395162171284, 6.68083200552010376625359042361, 7.82501926201173289483677548042, 8.581221952443711250433991524837, 9.772672548280872213754284291078, 10.64559793891441370105886069719

Graph of the $Z$-function along the critical line